How would you animate a particle system to start dispersed, converge to fill a mesh, then explode?
That is a job for keyed particles, you will most likely want to watch some tutorial videos explaining this in more detail.

Start by creating multiple particle systems, one for the starting position with the particles dispersed, a second one filling the shape that you want, then a third exploding them outwards. Then you setup another with physics set to keyed, in the keys list you add an entry for each system you have setup previously in the order you want them run. 

When played the particles will move from position to position of each particle system.

Here is a quick example file that may help.